diff --git "a/AdE4T4oBgHgl3EQf4w7g/content/tmp_files/load_file.txt" "b/AdE4T4oBgHgl3EQf4w7g/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/AdE4T4oBgHgl3EQf4w7g/content/tmp_files/load_file.txt" @@ -0,0 +1,1133 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf,len=1132 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='05317v1 [gr-qc] 12 Jan 2023 Next-to-leading-order Solution to Kerr-Newman Black Hole Superradiance Shou-Shan Bao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' ∗ Qi-Xuan Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' † and Hong Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' ‡ 1Institute of Frontier and Interdisciplinary Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Key Laboratory of Particle Physics and Particle Irradiation (MOE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Shandong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' QingDao 266237,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' China 2Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Blackett Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Imperial College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' SW7 2AZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' United Kingdom (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 2023) The superradiant instabilities of Kerr-Newman black holes with charged or uncharged massive spin-0 fields are calculated analytically to the next-to-leading order in the limit of α ∼ rgµ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A missing factor of 1/2 in the previous leading-order result is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The next-to-leading order result has a compact form and is in good agreement with existing numerical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The percentage error increases with α, from a few percent for α ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 to about 50% for α ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Massive neutral scalars too heavy to be produced with Kerr black hole superradiance may exist in the superradiant region of Kerr-Newman black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' INTRODUCTION Ultralight boson condensate could form around a rotat- ing black hole (BH) if the boson’s Compton wavelength is comparable to the size of the BH horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' With proper choice of parameters, such scalar condensate can con- tinuously extract energy and angular momentum from the BH until the BH spin is below some critical value and/or nonlinear effects become important[1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' This phenomenon is known as BH superradiance [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' There exist numerous works on various bosons, including spin- 0 [1, 6–26], spin-1 [25–38] and spin-2 [39, 40] fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this work, we focus on the ultralight scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The super- radiance of other types of bosons could be found in the comprehensive review [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The scalar superradiance, especially with a Kerr BH, is important in phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Such BH-condensate sys- tems have been widely studied for constraining the scalar properties and for the possible observation of the GW emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It has been shown that the BH evolves along the Regge trajectories on the mass-spin plot if the su- perradiant effect is strong [1, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Consequently, there are “holes” on the Regge plot in which BHs cannot re- side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Combing with the observed BH spin distribution, favored and unfavored scalar mass ranges can be iden- tified [42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' On the other hand, with the continuous GW generated by the BH-condensate, works has been done to study the possibility of resolving these systems from the backgrounds [1, 13–15, 18, 19, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The pos- itive frequency drift [13, 27] and the beat-like pattern [46] have been proposed to distinguish them from other monochromatic GW sources, such as neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The unresolved BH-condensate systems have also been care- fully studied as stochastic backgrounds for GW detectors [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The phenomenological study of BH superradiance de- pends on the accurate determination of the bound state’s ∗ ssbao@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='cn † qixuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='xu22@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='uk ‡ hong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='zhang@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='cn eigenfrequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For Kerr BHs, the numerical continued fraction method was first proposed by Leaver for mass- less scalars [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It is later developed for massive scalars in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [8] and further refined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the small α ∼ rgµ limit, an analytic approximation was obtained by Detweiler [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Nonetheless, these two solutions are not consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The problem is recently resolved in our previous work by including the next-to- leading-order (NLO) contribution to the analytic approx- imation [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A power-counting strategy is also proposed which facilitates the NLO calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [48], Damour et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' have shown that the su- perradiance can also be realized with a charged massive scalar field in Kerr-Newman spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Comparably, it does not attract as much attention as that for Kerr BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It may be because the Kerr-Newman BH (KNBH) is unlikely to play important roles in astrophysics [49– 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Nonetheless, as pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [52], the KNBH provides an ideal testing ground for studying the inter- play between gravity and electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the pre- vious studies of scalar superradiance with KNBHs, De- tweiler’s method has been applied to obtain the leading- order (LO) analytic approximation at the α ≪ 1 limit [53–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The numerical solution has also been achieved using the 3-term continued fraction method [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The pa- rameter space of the KNBH superradiance is also probed by analyzing the existence of the potential well [57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this work, we refine the power-counting strategy in our previous work and apply it to calculate the NLO con- tribution of the KNBH superradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A compact NLO expression for α ≪ 1 is obtained which could be straight- forwardly applied to phenomenological study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The scalar field can be either neutral or charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' By comparing to the existing numerical results, the percentage error of the analytic approximation increases with α, from a few percent for α ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 to about 50% for α ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' II, we briefly review the Klein-Gordon equation to be solved and ob- tain the superradiance condition from its solution at the outer horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Detweiler’s method is applied to derive the LO and NLO analytic expressions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' IV, the obtained analytic expressions are compared 2 to the existing numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Some effects rel- evant to phenomenology are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Finally, we summarize our results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' SCALARS IN KERR-NEWMAN SPACETIME The spacetime around a KNBH with mass M, angular momentum J and charge Q can be expressed in Boyer- Lindquist coordinates [60], ds2 = − � 1 − 2rgr − Q2 Σ2 � dt2 + Σ2 ∆ dr2 + Σ2dθ2 + � (r2 + a2) + (2rgr − Q2)a2 sin2 θ Σ2 � sin2 θdϕ2 − 2(2rgr − Q2)a sin2 θ Σ2 dtdϕ, (1) with a = J/M, (2a) rg = GM, (2b) Σ2 = r2 + a2 cos2 θ, (2c) ∆ = r2 − 2rgr + a2 + Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (2d) The equation ∆ = 0 gives two event horizons at r± = rg ± b with b = � r2g − a2 − Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this work, we only consider the KNBHs with r2 g − a2 − Q2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' To study the superradiance of a scalar field close to a BH, one needs to solve the combined Einstein and Klein- Gordon field equations, which is a very difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Es- pecially, the existence of the scalar perturbs the space- time around the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Nonetheless, it has been shown that this perturbation could be safely ignored due to the tiny energy-stress tensor of the scalar cloud for Kerr BH [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We assume the same situation happens for the KNBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We further assume the self-interaction of the scalar field can also be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Then the problem reduces to solv- ing the Klein-Gordon equation on the stationary Kerr- Newman background, (∇α − iqAα)(∇α − iqAα)φ − µ2φ = 0, (3) where µ and q are the mass and electric charge of the scalar field, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The vector Aα is the back- ground electromagnetic potential, Aα = Qr Σ2 � −1, 0, 0, a sin2 θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (4) For complex scalars, φ can be written with the separation of variables, φ(t, r, θ, ϕ) = � l,m � dωRlm(r)Slm(θ)eimϕe−iωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (5) Inserting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (3), one obtains the angular equa- tion, 1 sin θ d dθ � sin θdSlm dθ � + � −a2(µ2 − ω2) cos2 θ − m2 sin2 θ + Λlm � Slm = 0, (6) where Λlm is the eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Its solution Slm(θ) is called the spheroidal harmonic function, whose properties can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The corresponding radial equation is [62], ∆ d dr � ∆dRlm dr � + U(r)Rlm = 0, (7) with U(r) =[ω(a2 + r2) − am − qQr]2 + ∆[2amω − µ2r2 − a2ω2 − Λlm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (8) These are the equations for the complex scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For real scalars, one should set q = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (3) and choose only the real part on the right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The obtained angular and radial equations are the same as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (6) and (7) with q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the rest of this paper, we focus on the equations for the complex scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The case for the real scalars can then be simply obtained by choosing q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' To obtain a constraint of parameters that allow super- radiance, we change to the tortoise coordinates, dr∗ = r2 + a2 ∆ dr, (9) with which the interesting region r ∈ (r+, +∞) corre- sponds to r∗ ∈ (−∞, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We also define, R∗(r∗) = � r2 + a2R(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (10) Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (7) can be rewritten into a Schr¨odinger-like equation, d2R∗(r∗) dr2∗ − V (r)R∗(r∗) = 0, (11) where the effective potential is, V (r) = − � ω − am + qQr a2 + r2 �2 + ∆µ2 a2 + r2 − ∆ (a2 + r2)2 � 2amω − Λlm + a2(µ2 − ω2) � + ∆[∆ + 2r(r − rg)] (a2 + r2)3 − 3∆2r2 (a2 + r2)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (12) In the region close to the outer horizon r+, the potential has the asymptotic form, lim r→r+ V (r) = −(ω − ωc)2 + O(r − r+), (13) 3 where the critical frequency is defined as, ωc = ma + qQr+ r2 + + a2 = ma + qQr+ 2rgr+ − Q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (14) Inserting this asymptotic expression of V (r) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (11), one gets the solution at the outer horizon, lim r∗→−∞R∗(r∗) = d1e−i(ω−ωc)r∗ + d2ei(ω−ωc)r∗, (15) where the first term is the wave falling into the outer hori- zon, and the second term is the wave escaping from the outer horizon, d1 and d2 are their amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Physically, nothing can escape from the horizon, indicating d2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The superradiance requires the phase velocity and the group velocity to be in opposite directions, which leads to the superradiance condition for a KNBH, Re(ω) < ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (16) With Q fixed, it is more relaxed (strict) compared to the superradiance condition of a Kerr BH if the charges of the scalar and the BH have the same sign (different signs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' ANALYTIC SOLUTION AT α ≪ 1 In the small α limit, the asymptotic matching method first proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [6] gives a reasonable approxima- tion of the complex eigenfrequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In a previous work, we have further calculated the NLO contribution for Kerr BH superradiance [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The NLO result has a much bet- ter agreement with the numerical solutions compared to the LO approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the current work, we apply the method to KNBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this section, we first repeat the LO approximation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A missing factor of 1/2 is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We then continue to calculate the NLO contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The calculation is valid for both real and complex scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For a real scalar field, one simply sets q = 0 throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Leading-order Approximation We first formally introduce the power-counting param- eter α ∼ rgµ for the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The scaling of other pa- rameters are Re ω ∼ µ ∼ q and a ∼ Q ∼ r+ ∼ r− ∼ rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Unlike some previous calculations in which α is defined to be rgµ, here we leave α as a power-counting parame- ter, which could be rgµ or any other quantity with the same scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the limit r → +∞, the derivative term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (7) divided by ∆2 can be written into a familiar form, 1 ∆ d dr � ∆dR dr � ≈ d2R dr2 + 2 r dR dr = 1 r d2 dr2 (rR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (17) The second term on the left side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (7) divided by ∆2 can be expanded in powers of rg/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Keeping terms up to r2 g/r2, the radial function at large r limit (r ≫ rg) can be simplified as, d2 dr2 (rR) + � (ω2 − µ2) + 2(2rgω2 − rgµ2 − qQω) r − l′(l′ + 1) r2 + O(r−3) � rR = 0, (18) where l′(l′ + 1) =Λlm + 4r2 g(µ2 − 3ω2) + a2(ω2 − µ2) + Q2(2ω2 − q2 − µ2) + 8rgqQω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (19) The l′ is related to the orbital angular number by, l′ = l + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (20) Here ǫ ∼ O(α2) plays the role of a regulator and cannot be simply dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For a confined profile, the real part of ω is less than the boson mass µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The physical solution is the one that decays exponentially at large r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It is more convenient to define, κ = � µ2 − ω2, (21) λ = 2rgω2 − rgµ2 − qQω κ , (22) y = κr, (23) u(y) = yR �y κ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (24) Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (18) can be rewritten as, d2u(y) dy2 + � −1 + 2λ y − l′(l′ + 1) y2 � u(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (25) The two solutions are Whittaker functions, and only one of them has the correct behavior at r → +∞ required by the bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The solution with the correct behavior can be further written in terms of confluent hypergeo- metric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Finally, the radial function at large r 4 is, R(r) = e−κr(2κr)l′U(l′ + 1 − λ, 2l′ + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 2κr), (26) up to an arbitrary normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The bound states only exist if λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The super- radiance conditon in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (16) gives another constraint 2rgω < (ma + qQr+)/r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Combining these two inequal- ities, one can obtain, 0 < 2rgω2 − rgµ2 − qQω < �ma r+ + qQ � ω − rgµ2 − qQω = ma r+ ω − rgµ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (27) So there is no superradiant bound state if m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It also shows that Reissner-Nordstr¨om BHs could not hold bounded scalar clouds [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The minimum KNBH spin a allowing superradiant instability is approximately rgr+µ/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Next, we look at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (7) in the small r limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For BH superradiance, the inner boundary is the outer horizon r = r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It is more convenient to write the radial function in terms of z = (r − r+)/2b, z(z + 1) d dz � z(z + 1)dR dz � + U(z)R = 0, (28) where U(z) can be written as an expansion of z, U(z) = p2 + z �4rgr+ω b � r+ω − am 2r+ − Q2ω 2rg � − (Λlm + r2 +µ2 + a2ω2) + qQ b (am + r+qQ − a2ω − 3r2 +ω) � + z2(a2ω2 − Λlm + 2µ2a2 − 3µ2r2 + + 6r2 +ω2 + 2Q2µ2 + q2Q2 − 6r+qQω) + 4z3b [rgµ2 + 2r+(ω2 − µ2) − qQω] + 4z4b2(ω2 − µ2), (29) in which, p = (r2 + + a2) 2b (ω − ωc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (30) Note that both p and rgωc scale as O(α0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the limit of small α, the Λlm has the expanded form Λlm = l(l+1)+O(α4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' At the LO of α, we get the radial equation in limit (r − r+) ≪ max(1/ω, 1/µ), z(z + 1) d dz � z(z + 1)dR dz � + � p2 − l′(l′ + 1)z(1 + z) � R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (31) In principle, the l′ should be replaced by l in this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Nonetheless, the ǫ in l′ plays the role of a regulator in the intermediate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It will be set to zero at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The general solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (31) is a linear combination of two associated Legendre functions, and the physical solution is the one with the ingoing wave at r → r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' After changing the variable back to r, the solution of the radial function is, R(r) = �r − r+ r − r− �−ip 2F1 � −l′, l′ + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 1 − 2ip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' −r − r+ 2b � , (32) up to an arbitrary normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Next, we apply the matching method first proposed in [6] and further developed recently in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (26) is only valid in r ≫ rg limit, while the solution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (32) requires r ≪ rgα−2 from the ig- norance of terms proportional to z3 and z4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' They have an overlapped region in the limit α ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this region, the two solutions are expected to have the same behav- ior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The behavior of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (26) in the overlapped region is obtained by looking at its small r limit, which is, (2κ)l′Γ(−2l′ − 1) Γ(−l′ − λ) rl′ + (2κ)−l′−1Γ(2l′ + 1) Γ(l′ + 1 − λ) r−l′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (33) On the other hand, the behavior of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (32) in the over- lapped region is obtained by looking at its large r limit, which is, (2b)−l′Γ(2l′ + 1) Γ(l′ + 1)Γ(l′ + 1 − 2ip)rl′ + (2b)l′+1Γ(−2l′ − 1) Γ(−l′ − 2ip)Γ(−l′) r−l′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (34) The ratio of the coefficients of the rl′ and r−l′−1 should be the same for the two solutions in the overlap region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The obtained equation is the eigenequation of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It can be 5 solved perturbatively by the observation that the second term in the expression (33) must be suppressed at small r, indicating l′+1−λ is very close to zero or some negative integer, l′ + 1 − λ = −n − δλ, (35) where |δλ| ≪ 1 and n is zero or a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Following the convention in literature, we also define ¯n = n + l + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Then the above relation is re-expressed as λ = ¯n + ǫ + δλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' At LO of α, it reduces to λ = ¯n + δλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Combining with the definition of λ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (22), the rgκ scales as α2, which is important in power-counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Since |δλ| ≪ 1, one could solve for δλ perturbatively with ex- pressions (33) and (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The LO calculation of δλ for Kerr BHs was completed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [6], with the regulator ǫ set to zero from the be- ginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Recently, we have confirmed a missing factor of 1/2 in that result [21], which was first identified in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The missing factor is conjectured to be from mistreatments of Γ functions with negative integer argu- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The correct formula is provided in the appendix of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' This subtle calculation turns out to be straight- forward with the regulator ǫ kept in the intermediate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' More details could be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For KNBHs, the first LO calculation of δλ was completed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It followed the same steps in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [6] and missed the factor 1/2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' After the correction, the LO result of δλ is, δλ(0) = − ip (4κb)2l+1 (n + 2l + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='(l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' )2 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [(2l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (2l + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' ]2 l� j=1 (j2 + 4p2), (36) where the superscript (0) indicates that it is the LO re- sult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It scales as O(α4l+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The eigen-frequency ω can be expressed in terms of δλ with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (22) and (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Defining ω = ω0 + ω1δλ(0) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (22) and expanding it to the linear term of δλ(0), one arrives at, λ = rg(2ω2 0 − µ2) − qQω0 � µ2 − ω2 0 + rgω0ω1(3µ2 − 2ω2 0) − qQµ2ω1 (µ2 − ω2 0)3/2 δλ(0) + O � (δλ(0))2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (37) On the other hand, we have λ = ¯n + δλ(0) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Then it is straightforward to get, rg(2ω2 0 − µ2) − qQω0 � µ2 − ω2 0 = ¯n, (38a) rgω0ω1(3µ2 − 2ω2 0) − qQµ2ω1 (µ2 − ω2 0)3/2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (38b) Note that in getting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (38a), we have ignored the ǫ which could be traced back to the l′ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' This omission leads to an error in rgω0 at the order of O(α5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Solving ω0 perturbatively from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (38a), one arrives at, ω(0) 0 µ = 1 − 1 2 �rgµ − qQ ¯n �2 + O(α4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (39) Then the ω1 could be expressed in terms of ω0 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (38b) and expanded in powers of α, ω(0) 1 µ = (rgµ − qQ)2 ¯n3 + O(α4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (40) Since both ω0 and ω1 are real, ω0 and ω1δλ(0) are the leading terms of the real and imaginary parts of ω, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Especially, the imaginary part of ω scales as O(α4l+5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Next-to-leading-order Approximation In a previous work, we have carefully studied the su- perradiance of a real scalar field around a Kerr BH [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The LO eigenfrequency ω obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [6] has an er- ror as large as 160% compared to the numerical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' After correcting the missing factor 1/2, the convergence is improved, with the error ≲ 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Except for the large discrepancy, the LO result also has some strange behav- iors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Since the LO result is the leading term in the Taylor series of the exact ω at α = 0, it is expected to converge to the exact ω with α approaching zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Nonetheless, the relative error seems to be a nonzero constant for small α, reaching as large as 30% at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='07 for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' This discrepancy at small α puts the question on the power- counting strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Moreover, the discrepancy at small α increases quickly with the BH spin parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' These problems are solved by adding the NLO correc- tion of ω [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Below we follow the same steps for the KNBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The key observation is that the first term in the square bracket in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (29), which scales as α2, is enhanced by a factor of 1/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For BHs with large spin a and/or charge Q, this term can be as important as the LO contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Other NLO contributions are also added for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The first NLO correction appears as ǫ in the asymp- totic radial wave function at large r, which is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It can be calculated from the definition of l′ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (19), ǫ = −8r2 gµ2 + Q2µ2 + 8rgqQµ − q2Q2 2l + 1 + O(α4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (41) The second NLO contribution is from the asymptotic ra- dial wave function at small r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The potential U(z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (29) can be approximated by p2−l′(l′+1)z(1+z)+zd, where d is defined as d =(4rgµ − 2qQ)p − 2(4rg − r+)rgµ2 + 2µqQ(4rg − r+) − q2Q2 + O(α3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 6 Up to an arbitrary normalization, the corresponding ra- dial function at the NLO is, R(r) =(r − r−) √ d−p2 (r − r+)ip 2F1 � − l′ − ip + � d − p2, l′ + 1 − ip + � d − p2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 1 − 2ip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' −r − r+ 2b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (42) In the r → +∞ limit, the asymptotic behavior of this function is, (2b)−l′−ip+√ d−p2Γ(2l′ + 1)Γ(1 − 2ip) Γ(l′ + 1 − ip − � d − p2)Γ(l′ + 1 − ip + � d − p2) rl′ + (2b)l′+1−ip+√ d−p2Γ(−2l′ − 1)Γ(1 − 2ip) Γ(−l′ − ip − � d − p2)Γ(−l′ − ip + � d − p2) r−l′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (43) Following similar matching steps above, the NLO contri- bution of δλ could be obtained after some algebra, δλ(1) = � d 2ǫ − ǫ 2 − ip � (4κb)2l′+1 Γ(n + 2l′ + 2)Γpd n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [Γ(2l′ + 1)Γ(2l′ + 2)]2 , (44) where the superscript (1) indicates it is the NLO result, and the Γpd is defined as, Γpd = ���Γ(l′ + 1 + ip + � d − p2)Γ(l′ + 1 + ip − � d − p2) ��� 2 Γ(1 + 2ǫ)Γ(1 − 2ǫ) Γ(1 − ip − � d − p2 − ǫ)Γ(1 + ip + � d − p2 + ǫ)Γ(1 − ip + � d − p2 − ǫ)Γ(1 + ip − � d − p2 + ǫ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (45) The last NLO contribution is from ω0 and ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Defining ω = ω(1) 0 + ω(1) 1 δλ(1), the expansion of λ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (37) is still valid, only with δλ(0) replaced by δλ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Combining with λ = ¯n + ǫ + δλ(1), one could follow the same steps as in the LO calculation and obtain, ω(1) 0 µ =1 − 1 2 �rgµ − qQ ¯n �2 + (rgµ − qQ)2 8¯n4 [3(rgµ − qQ)(5rgµ − qQ) + 8¯nǫ] + O(α6), (46a) ω(1) 1 µ =(rgµ − qQ)2 ¯n3 − 3(rgµ − qQ)2 2¯n5 [(rgµ − qQ)(5rgµ − qQ) + 2¯nǫ] + O(α6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (46b) Finally, we discuss a subtle problem related to the ω dependence in the definition of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the calculation of the δλ(1), the ω in p should be replaced by ω(0) 0 , rather than ω(1) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Here we explain the reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In deriving the small-r asymptotic form of the radial function, we approximate U(z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (29) by p2 − l′(l′ + 1)z(z + z) + zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The coefficient of z and z2 are accurate at O(α2) and O(α0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' At z ∼ O(α), this two terms are at the same order of O(α4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Consequently, we only need to keep the terms in p2 up to O(α4), which then leads to ω = ω(0) 0 in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In comparison to the numerical calculation, this choice of ω gives a satisfactory NLO result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Using ω(1) 0 in p is not as satisfactory, due to partially including the high-order contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' RESULTS The eigenfrequency of the Kerr BH superradiance has been studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [8, 10, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In comparison, the case for Kerr-Newman BH has two more parameters, the BH charge Q and the scalar charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this section, we first study the superradiance of a neutral scalar field, focusing on the effect of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Then we consider the superradiance of a charged scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Comparisons with the numerical calculations in the literature are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Neutral Scalar Fields In the following study of neutral scalar superradiance, we adopt the NLO δλ(1) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (44), where the scalar charge q is set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The ω(1) 0 and ω(1) 1 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (46) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Then the NLO eigen-frequency is ω = ω(1) 0 + ω(1) 1 δλ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The BH charge Q cannot be chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In our derivation, we have implicitly assumed the KNBH has 7 horizons, which requires |Q| ≤ � r2g − a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In addition, neutral scalars could not distinguish the sign of the BH charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Mathematically, it means the BH charge Q can only appear in the formulas as Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' So it is sufficient to only consider positive Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The superradiance condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (16) with q = 0 has the same form as the Kerr BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The effect of the BH charge Q is hidden in r+ = rg + � r2g − a2 − Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Keeping the BH mass M and spin a fixed, larger charge Q results in a larger upper limit of Re(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Thus massive scalars too heavy to be produced with Kerr BH superradiance may exist in the superradiant region of KNBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Q=0 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 10-13 10-11 10-9 10-7 rg � Im ( � ) / � l=m=1 l=m=2 l=m=3 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 Q=0 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='6 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 10-13 10-11 10-9 10-7 rg � Im ( � ) / � l=m=1 l=m=2 l=m=3 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The imaginary part of NLO eigenfrequency with q = 0 as a function of rgµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Only the curves with n = 0 are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the top (bottom) panel, the BH spin a is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In both panels, from left to right, the three bunches correspond to l = m = 1, 2, 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In each bunch, the curves with different colors correspond to different values of the BH charge Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 1 shows the imaginary part of ω as a function of rgµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For comparison, the curves for Kerr BHs are also shown, labeled with Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' All curves have the same qualitative behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' With an increasing value of rgµ, they first increase, then drop rapidly to below zero after reaching the maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' There are three effects of the BH charge Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Firstly, the superradiant region of rgµ is enlarged with larger Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Correspondingly, the peak of Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3 rg μ s (Q) a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='6 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 rg μ s (Q) a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Factor s(Q) with q = 0 as a function of rgµ for BH spin a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 (upper panel) and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7 (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The vertical dashed line in each panel labels the value of rgµ where Im ω(Q = 0) reaches its maximum value for the corresponding spin parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' the curve moves to the right with increasing Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The maximum rgµ with positive Im(ω) is quite accurately determined by µ = ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Secondly, the maximum Im(ω) increases with larger Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Fixing the BH spin to be a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9, the maximum values of rgIm(ω) with Q = 0 are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='088 × 10−8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='427 × 10−9 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='029 × 10−10 for l = m = 1, 2, 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The numbers for Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='43 are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='476 × 10−7, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='006 × 10−8 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='760 × 10−10, which are larger than the Q = 0 cases by factors of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='07, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='26 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For BHs with spin a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7, the maximum Q is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The enhancement factors are 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='29, 269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='91, and 707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='16, for l = m = 1, 2, 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Finally, in the ranges of small rgµ before reaching the round peaks of the Q = 0 curves, the charge Q turns out to impede the growth of the scalar clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We define a factor s(Q) as s(Q) = Im ω(Q) Im ω(Q = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (47) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 2, we show s(Q) as a function of rgµ, for two different BH spins and several values of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Interestingly, the suppression factor varies slowly with rgµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It decreases with increasing Q, reaching the minimum value ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='8 for 8 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [56], the authors claim that when a ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='997rg, the maximum value of Im ω decreases as Q grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We do not observe the same behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For any spin parameter a, the peak value of Im ω from the NLO approximation increases monotonically with Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Charged Scalar Fields In this part, we study the superradiance of KNBHs under charged scalar perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The NLO eigenfre- qency is given by ω = ω(1) 0 + ω(1) 1 δλ(1), with the NLO δλ(1) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (44), and the ω(1) 0 and ω(1) 1 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Note that the ω in p should take the form of ω(0) 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (39), as explained at the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We also compare the NLO results to the LO ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The latter is given by ω = ω(0) 0 + ω(0) 1 δλ(0), with the expressions de- fined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (36), (39) and (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The ω in p is replaced by µ for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 60 40 20 0 20 10-9 10-8 10-7 q Im (ω) μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3 μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='41 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Comparison of the numerical result and the analytic approximations for n = 0, l = m = 1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='98, and Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01, with rg chosen to be 1 for compacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The imaginary part of ω is plotted as a function of the scalar field charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The dashed (solid) curves are the LO (NLO) approximations and the scattered dots are numerical results taken from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 6 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The curves with different colors correspond to different values of µ, labeled above the corresponding curves with the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 3 shows the comparison of the LO and NLO ap- proximations to the numerical results taken from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 6 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The NLO approximation agrees much better with the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In particular, the average per- centage errors of the NLO results for the points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 3 are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7%, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9%, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7% and 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3% for rgµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='41, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' These numbers can be used as estimates of the NLO results for different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Moreover, the convergence of NLO results is better for a smaller value of rgµ, qualifying the power-counting strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' To the contrary, the LO results do not seem to con- verge to the numerical result at small rgµ, which is also observed for Kerr BHs [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The reason for the bad con- vergence of the LO result is explained at the beginning of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A caveat is that the curves for the LO approx- imations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 3 are not the same as those in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The latter misses a factor of 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' I shows the comparison of the NLO results and the numerical solutions for five more parameter sets in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' They are the most unstable modes with different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The percentage uncertainty of the NLO approximation varies from 14% to 29% compared to the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Comparison of the NLO approximations of Im(ω) with the numerical results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [56] (cases A to D) and from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [53] (case E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' All cases are with n = 0 and l = m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The numbers below assume rg = 1 for compacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The percentage error is calculated by taking the difference between the approximation and the numerical result, then dividing it by the numerical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Case A: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9, Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 , q = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='264, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='282;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Case B: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='99, Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1105 , q = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='6335, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='397;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Case C: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='997, Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='004 , q = −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='91, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='39822;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Case D: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='997, Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0001, q = −756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='68, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='39816;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Case E: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='98, Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01, q = −8, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Case Type Im(ω) % error LO 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='623×10−9 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9% A NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='882×10−8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5% Numerical 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='243×10−8 LO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='224×10−8 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9% B NLO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='981×10−7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1% Numerical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='736×10−7 LO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='264×10−8 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9% C NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='041×10−7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1% Numerical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='788×10−7 LO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='263×10−8 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9% D NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='041×10−7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1% Numerical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='788×10−7 LO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='27×10−8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='8% E NLO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='39×10−7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7% Numerical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='13×10−7 Next, we analyze the effect of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the formulas, the q and Q appears as qQ and Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' So it is sufficient to consider the case with Q > 0, and with q being either positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' There are two constraints for the existence of the superradiant bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The superra- diance requires ω < ωc in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' And the existence of the bound states gives the second constraint λ > 0 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (22), which is approximately rgµ − qQ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' If the scalar and the KNBH at the center have opposite charges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' qQ < 0, the scalar cloud is more tightly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this case, the second constraint above is automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 4 shows the imaginary part of ω as a function of rgµ in the n = 0, l = m = 1 bound state, with BH spin a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 and charge Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The scalar charge q varies from −45 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The region of superradiance shrinks when q is more negative, which is a consequence that ωc decreases with q for fixed 9 q=0 q=-5 q=-10 q=-15 q=-30 q=-45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='35 10-20 10-17 10-14 10-11 10-8 rg μ Im ( � ) / μ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The imaginary part of NLO eigenfrequency as a func- tion of rgµ with different negative values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Other param- eters are n = 0, l = m = 1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 and Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The maximum value of Im(ω) obtained by varying q, with a and Q fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (a,Q) q Im(ω) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10313×10−8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10329×10−8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10329×10−8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10268×10−8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10814×10−8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='02) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10831×10−8 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10815×10−8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10682×10−8 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14247×10−10 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14270×10−10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14260×10−10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14104×10−10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14863×10−10 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='02) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14888×10−10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14726×10−10 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='13927×10−10 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The peak value of Im(ω) seems to be smaller with decreasing q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Nonetheless, a more careful study shows that the maximum Im(ω) happens at some small but nonzero |q| (see Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' If the charges of the scalar and the KNBH have the same sign, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' qQ > 0, the scalar cloud is less bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The second constraint above gives rgµ > qQ for the ex- istence of bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 5 shows the imaginary part of ω as a function of rgµ in the n = 0, l = m = 1 bound state, with BH spin a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 and charge Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' With larger value of positive q, the superradiance region shrinks and the peak is lower as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' q=0 q=5 q=10 q=15 q=30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3 0� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 10-20 10-17 10-14 10-11 10-8 rg μ Im ( � ) / μ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The imaginary part of NLO eigenfrequency as a func- tion of rgµ with different positive values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Other parame- ters are n = 0, l = m = 1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 and Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' CONCLUSION In this work, we have studied the scalar superradi- ant instability of the KNBH and obtained the LO and NLO expressions of the superradiant rate in the regime of α ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The calculation is based on the matching method which is proposed by Detweiler for Kerr BHs in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [6] and developed in our previous work [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this manuscript, we further refine the power-counting strat- egy and apply it to the KNBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The LO scalar superradiant rate for KNBH has been calculated previously in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' With our refined power-counting strategy, a similar result is obtained but with an extra overall factor of 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We conjecture the factor is from the mistreatment of the Γ functions with negative integer arguments, similar to the case of Kerr BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' More analysis could be found in our previous work [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We compare the LO and NLO results with the existing numerical calculations in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The LO results are smaller than the numerical solutions by an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' To the contrary, the percentage error of the NLO result ranges from a few percent to about 50%, de- pending on the value of α (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 3 and Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In particular, the error of the NLO result decreases for a smaller value of α, qualifying our power-counting strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The obtained NLO expression has a compact form and can be straightforwardly applied to phenomenolog- ical studies of the KNBH superradiance as well as the ultralight scalars, either neutral or charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Besides the superradiance condition Re(ω) < mΩH as the Kerr BHs, there is another condition rgµ > qQ for the existence of bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For neutral scalars, larger BH charge Q leads to a larger superradiant range of rgµ as well as the maximum superradiant rate (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Thus mas- sive neutral scalars too heavy to be produced with Kerr 10 BH superradiance may exist in the superradiant region of KNBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The situation is different for charged scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For fixed BH spin a and charge Q, increasing the scalar charge q always leads to narrower superradiant range of rgµ (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Interestingly, the maximum su- perradiant rate happens at a small negative scalar charge q (see Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We have no explanation for this obser- vation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work is supported in part by the National Nat- ural Science Foundation of China (NSFC) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 12075136 and the Natural Science Foundation of Shandong Province under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' ZR2020MA094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Arvanitaki and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Dubovsky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' D 83, 044026 (2011) [arXiv:1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3558 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Yoshino and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Kodama, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2, 023514 (2013) [arXiv:1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='6725 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [40] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Brito, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Grillo and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Pani, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 124, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='21, 211101 (2020) [arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='04055 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [41] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Brito, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Cardoso and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Pani, Physics,” Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Notes Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 906, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1-237 (2015) 2020, ISBN 978-3- 319-18999-4, 978-3-319-19000-6, 978-3-030-46621-3, 978- 3-030-46622-0 [arXiv:1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='06570 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [42] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Ng, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Hannuksela, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Vitale and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Li, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' D 103, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='6, 063010 (2021) [arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} 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